A closer look at the hysteresis loop for ferromagnets · of the hysteresis loop for ferromagnets...
Transcript of A closer look at the hysteresis loop for ferromagnets · of the hysteresis loop for ferromagnets...
A closer look at the hysteresis loop for ferromagnets - A survey of misconceptions and misinterpretations in textbooks
Hilda W. F. Sung and Czeslaw Rudowicz
Department of Physics and Materials Science, City University of Hong Kong, 83 Tat Chee Avenue,
Kowloon, Hong Kong SAR, People�s Republic of China
This article describes various misconceptions and misinterpretations concerning presentation
of the hysteresis loop for ferromagnets occurring in undergraduate textbooks. These problems
originate from our teaching a solid state / condensed matter physics (SSP/CMP) course. A closer
look at the definition of the 'coercivity' reveals two distinct notions referred to the hysteresis
loop: B vs H or M vs H, which can be easily confused and, in fact, are confused in several
textbooks. The properties of the M vs H type hysteresis loop are often ascribed to the B vs H
type loops, giving rise to various misconceptions. An extensive survey of textbooks at first in the
SSP/CMP area and later extended into the areas of general physics, materials science and
magnetism / electromagnetism has been carried out. Relevant encyclopedias and physics
dictionaries have also been consulted. The survey has revealed various other substantial
misconceptions and/or misinterpretations than those originally identified in the SSP/CMP area.
The results are presented here to help clarifying the misconceptions and misinterpretations in
question. The physics education aspects arising from the textbook survey are also discussed.
Additionally, analysis of the CMP examination results concerning questions pertinent for the
hysteresis loop is provided.
Keywords: ferromagnetic materials; hysteresis loop; coercivity; remanence; saturation induction
1. Introduction
During years of teaching the solid state physics
(SSP), which more recently become the condensed
matter physics (CMP) course, one of us (CZR),
prompted by questions from curious students
(among others, HWFS), has realized that textbooks
contain often not only common misprints but
sometimes more serious misconceptions. The latter
occur mostly when the authors attempt to present a
more advanced topic in a simpler way using
schematic diagrams. One such case concerns
presentation of the magnetic hysteresis loop for
ferromagnetic materials. Having identified some
misconceptions existing in several textbooks
currently being used for our SSP/CMP course at
CityU, we have embarked on an extensive literature
survey. Search of physics education journals have
revealed only a few articles dealing with magnetism,
e.g. Hickey & Schibeci (1999), Hoon & Tanner
(1985). Interestingly, a review of middle school
physical science texts by Hubisz (http://www.psrc-
online.org/curriculum/book.html), which has
recently come to our attention, provides ample
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examples of various errors and misconceptions
together with pertinent critical comments. However,
none of these sources have provided clarifications of
the problems in question. To find out the extent of
these misconceptions existing in other physics areas,
we have surveyed a large number of available
textbooks pertinent for solid state / condensed
matter, general physics, materials science, and
magnetism / electromagnetism. Several pertinent
encyclopedias and physics dictionaries have also
been consulted. The survey has given us more than
we bargained for, namely, it has revealed various
other substantial misconceptions than those
originally identified in the SSP/CMP area. The
results of this survey are presented here for the
benefit of physics teachers (as well as researchers)
and students. The textbooks, in which no relevant
misconceptions and/or confusions were identified,
are not quoted in text, however, they are listed for
completeness in Appendix I in order to provide a
comprehensive information on the scope of our
survey.
In order to provide the counterexamples for the
misconceptions identified in the textbooks, we have
reviewed a sample of recent scientific journals
searching for real examples of the magnetic
hysteresis loop, beyond the schematic diagrams
found in most textbooks. To our surprise a number
of general misconceptions concerning magnetism
have been identified in this review. The results of
this review are presented in a separate article, which
focuses on the research aspects and provides recent
literature data on soft and hard magnetic materials
(Sung & Rudowicz, 2002; hereafter referred to as S
& R, 2002).
The root of the problem appears to be the
existence of two ways of presenting the hysteresis
loop for ferromagnets: (i) B vs H curve or (ii) M vs
H curve. In both cases, the �coercivity� (�coercive
force�) is defined as the point on the negative H-
axis, often using an identical symbol, most
commonly Hc. Yet it turns out that the two
meanings of �coercivity� are not equivalent. In some
textbooks the second notion of coercivity (M vs H)
is distinguished from the first one (B vs H) as the
�intrinsic� coercivity Hci. An apparent identification
of the two meanings of coercivity Hc (B vs H) and
Hci (M vs H) as well as of the properties of soft and
hard magnetic materials have lead to
misinterpretation of Hc as the point on the B vs H
hysteresis loop where the magnetization is zero.
This is evident, for example, in the statements
referring to Hc as the point at which �the sample is
again unmagnetized� (Serway, 1990) or �the field
required to demagnetize the sample� (Rogalski &
Palmer, 2000). Other misconceptions identified in
our textbook survey concern: 'saturation induction
Bsat� and the inclination of the B vs H curve after
saturation, shape of the hysteresis loop for soft
magnetic materials, and presentation of the
hysteresis loop for both soft and hard ferromagnets
in the same diagram. Minor problems concerning
terminology and the drawbacks of using schematic
diagrams are also discussed. Analysis of the
condensed matter physics examination results
concerning questions pertinent for the hysteresis
loop is provided to illustrate some popular
misconceptions in students' understanding.
2. Two notions of coercivity
For a ferromagnetic material, the magnetic
induction (or the magnetic field intensity) inside the
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sample, B, is defined as (see, e.g. any of the books
listed in References):
B = H + 4πM (CGS);
B = µo ( H + M ) (SI) (1)
where M is the magnetization induced inside the
sample by the applied magnetic field H. In the free
space: M = 0 and then in the SI units: B ≡ µo H,
where µo is the permeability of free space (µo = 4π x
10-7 [m kg A-2 sec-2]; note that the units [Hm-1] and
[WbA-1m-1] are also in use). The standard SI units
are: B [tesla] = [T], H and M [A/m], whereas B
[Gauss] = [G], H [Oersted] = [Oe], and M [emu/cc]
(see, e.g. Jiles, 1991; Anderson, 1989). Both the
CGS units and the SI units are provided since the
CGS unit system is in use in some textbooks
surveyed and comparisons of values need to be
made later.
In Fig. 1 we present schematically the hysteresis
curves for a ferromagnetic material together with
the definitions of the terms important for
technological applications of magnetic materials.
The two meanings of �coercivity� Hci and Hc as
defined on the diagrams: (a) the magnetization M vs
applied field H and (b) magnetic induction (or flux
density) B vs H, respectively, are clearly
distinguished. Both curves have a similar general
characteristic, except for one crucial point. After the
saturation point is reached, the M curve becomes a
straight line with exactly zero slope, whereas the
slope of the B curve reflects the constant magnetic
susceptibility and depends on the scale and units
used to plot B vs H (see below). In other words, the
B vs H curve does not saturate by approaching a
Fig. 1. Hysteresis curves for a ferromagnetic material: (a) M vs H: Mr is the remanent magnetization at H = 0;
Hci is the intrinsic coercivity, i.e. the reverse field that reduces M to zero; Ms is the saturation
magnetization; (b) B vs H: Br is the remanent induction (or �remanence�) at H = 0; Hc is the coercivity,
i.e. the reverse field required to reduce B to zero (adapted from Elliot, 1998).
Magnetic induction B = µo(H + M)
(b) (a)
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limiting value as in the case of the M vs H curve.
For an initially unmagnetized sample, i.e. M ≡ 0
at H = 0, as H increases from zero, M and B
increases as shown by the dashed curves in Fig. 1
(a) and (b), respectively. This magnetization process
is due to the motion and growth of the magnetic
domains, i.e. the areas with the same direction of
the local magnetization. For a full discussion of the
formation of hysteresis loop and the nature of
magnetic domains inside a ferromagnetic sample
one may refer to the specialized textbooks listed in
the References, e.g. Kittel (1996), Elliott (1998),
Dalven (1990), Skomski & Coey (1999). Here we
provide only a brief description of these aspects. A
distinction must be made at this point between the
magnetically isotropic materials, for which the
magnetization process does not depend on the
orientation of the sample in the applied field H, and
the anisotropic ones, which are magnetized first in
the easy direction at the lower values of H. In the
former case, as each domain magnetization tends
to rotate to the direction of the applied field Kittel
(1996), the domain wall displacements occur,
resulting in the growth of the volume of domains
favorably oriented (i.e. parallel) to the applied field
and the decrease of the unfavorably oriented
domains Kittel (1996). In the latter case, only after
the magnetic anisotropy (for definition, see, e.g.
Kittel (1996), Elliott (1998), Dalven (1990),
Skomski & Coey (1999), Jiles (1991)) is overcome
the sample is fully magnetized with the direction of
M along H. In either case, when this �saturation
point� is reached, the magnetization curve no longer
retraces the original dashed curve when H is
reduced. This is due to the irreversibility of the
domain wall displacements. When the applied field
H reaches again zero, the sample still retains some
magnetization due to the existence of domains still
aligned in the original direction of the applied field
Dalven (1990). The respective values at H = 0 are
defined (see, e.g. Kittel (1996), Elliott (1998),
Dalven (1990), Skomski & Coey (1999), Jiles
(1991)) as the remnant magnetization Mr, Fig. 1
(a), and the remnant induction Br, Fig. 1 (b). To
reduce the magnetization M and magnetic induction
B to zero, a reverse field is required known as the
coercive force or coercivity. The soft and hard
magnetic materials are distinguished by their small
and large area of the hysteresis loop, respectively.
By definition, the coercive force (coercivity)
defined in Fig. 1 (a), and that in Fig. 1 (b) are two
different notions, although their values may be very
close for some materials. In order to distinguish
them, some authors define either the related
coercivity (Kittel, 1996) or the intrinsic coercivity
(Elliott, 1998; Jiles, 1991) Hci as the reverse field
required to reduce the magnetization M from the
remnant magnetization Mr again to zero as shown in
Fig. 1 (a), whereas reserve the symbol Hc and the
name coercivity (coercive force) to denote the
reverse field required to reduce the magnetic
induction in the sample B to zero as shown in Fig. 1
(b), as done, e.g. by Kittel (1996). Hence, the
confusion between the two notions of coercivity
referred to the curve B vs H and the curve M vs H
can be avoided. Since a clear distinction between Hc
and Hci, is often not the case in a number of
textbooks, a question arises under what conditions
and for which magnetic systems, if any, Hc and Hci
can be considered as equivalent quantities. If it was
the case, the point �Hc on the B vs H curve would
also correspond to the magnetization M ≡ 0 as in the
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case of Hci on the M vs H curve. Only in one of the
books surveyed such approximation is explicitly
considered. Dalven (1990) shows that, in general,
the values of B and M are much larger than H in
both curves in Fig. 1. Hence, if H can be neglected
in Eq. 1, then B ≈ µoM. This turns to be valid only
for low values of H and the narrow hysteresis loop
pertinent for the soft magnetic materials. In other
words, the value of Hc and Hci are indeed very
close, so not identical, for the soft magnetic
materials only. In this case Hci in Fig. 1 (a) and
Hc in Fig. 1 (b) can be considered as two equivalent
points and hence M ≈ 0 at Hc as well.
The real examples of the magnetic hysteresis
loop, identified in our review (S & R, 2002) of a
sample of recent scientific journals, indicate that Hc
and Hci turn out to be significantly non-equivalent
for the hard magnetic materials. In the article (S &
R, 2002) we have also complied values of Hci, Hc,
and Br for several commercially available
permanent magnetic materials revealed by our
recent Internet search. These data indicate that
although Hc and Hci are of the same order of
magnitude, in a number of cases Hci is substantially
larger than Hc. Hence, in general, it is necessary to
distinguish between Hc and Hci. Moreover, as a
consequence of Hci ≠ Hc, the magnetization does not
reach zero at the point �Hc on the B vs H curve but
at a larger value of Hci indicated schematically in
Fig. 1 (b). However, in the early investigations of
magnetic materials, before the present day very
strong permanent magnets become available, the
values of Hc and Hci were in most cases not
distinguishable. As the advances in the magnet
technology progressed, more and more hard
magnetic materials have been developed, for which
the distinction between Hci and Hc is quite
pronounced (see Table 1 in S & R (2002)). The
presentation in most textbooks reflects the time lag
it takes for new materials or ideas to filter from
scientific journals into the textbooks as
'schematically presented established knowledge'.
3. Results of textbooks survey
In our survey of the presentation of the
hysteresis loop for ferromagnetic materials, in total
about 300 textbooks in the area of solid state /
condensed matter, general physics, materials
science, magnetism / electromagnetism as well as
several encyclopedias and physics dictionaries
available in City University library were examined.
We have identified around 130 books dealing with
the hysteresis loop. In order to save the space an
additional list of the books surveyed (37 items),
which deal with the hysteresis loop in a correct way
but are not quoted in the References, is available
from the authors upon request.
It appears that from the points of view under
investigation, generally, the encyclopedias and
physics dictionaries contain no explicit
misconceptions. This is mainly due to the fact that
the hysteresis loop is usually presented at a rather
low level of sophistication (see, e.g. Lapedes
(1978), Lord (1986), Meyers (1990), BesanHon
(1985), Parker (1993)). However, in a few instances
in the same source book both types of hysteresis
loop (B vs H and M vs H) are discussed in separate
articles written by different authors without
clarifying the distinct notions, which may also lead
to confusion. Examples include, e.g. (a) Anderson
& Blotzer (1999) and Vermariën et al (1999), and
(b) Arrott (1983), Donoho (1983), and Rhyne
(1983). Hence, these authoritative sources could not
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help us to clarify the intricacies we have
encountered. This have been achieved by consulting
more advanced books on the topic, e.g., Kittel
(1996), Dalven (1990), Skomski and Coey (1999),
and/or regular scientific journals (for references,
see, S & R, 2002).
Only a small number of books surveyed contain
both types of the curves: B vs H and M vs H as well
as provide clarification of the terminology
concerning Hc and Hci - Kittel (1996), Elliott (1998),
Dalven (1990), Skomski & Coey (1999), Jiles
(1991), Arrott (1983), Donoho (1983), Rhyne
(1983), Levy (1968), Anderson & Blotzer (1999),
Vermariën et al. (1999). Barger & Olsson (1987)
provide both graphs but terminology is only referred
to the B vs H graph. Most books deal only with one
type of the hysteresis loop. The B vs H curve,
which is more prone to misinterpretations, has been
used more often in the surveyed books in all areas.
A few books deal with the M vs H curve and
provide, with a few exceptions (see Section C
below), correct description and graphs (see, e.g.
Lovell et al , 1981; Aharoni, 1996; Wert &
Thomson, 1970; Elwell & Pointon, 1979). On the
other hand, the M vs H curve is dominant in
research papers surveyed (S & R, 2002).
Surprisingly, while most of the textbooks surveyed
attempt to adhere to the SI units, all but a few
research articles reviewed still use the CGS units.
This in itself is a worrying factor (S & R, 2002).
The various misconceptions and/or
misinterpretations identified in the course of our
comprehensive survey of textbooks can be classified
into five categories. Below we provide a systematic
review of the books with respect to the problems in
each category.
A. Misinterpretation of the coercivity Hc on the B
vs H curve as the point at which M=0.
This was the original problem which has
triggered the textbook survey. Various examples of
this misinterpretation, consisting in ascribing �zero
magnetization� to the point �Hc on the B vs H
hysteresis loop, are listed below with the nature of
the problem indicated by the pertinent sentences
quoted.
Solid state / condensed matter physics books
• �The magnetic field has to be reversed and
raised to a value Hc (called the coercive
force) in order to push domain walls over the
barriers so that we regain zero
magnetization.� (Wilson, 1979)
• �The point at which B=0 is the coercive field
and is usually designated as Hc. It represents
the magnetic field required to demagnetize the
specimen.� (Pollock, 1990)
• �The reverse field required to demagnetize
the material is called the coercive force, Hc.�
(Pollock, 1985)
• �To remove all magnetization from a
specimen then requires the application of a
field in the opposite direction termed the
coercive field.� (Elliott & Gibson, 1978)
• �H at c is called the coercive force and is a
measure of the field required to demagnetize
the sample.� (Rogalski & Palmer, 2000)
General physics books
• �The coercive force is a measure of the
magnitude of the external field in the opposite
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direction needed to reduce the residual
magnetization to zero.� (Ouseph, 1986)
• �In order to demagnetize the rod completely,
H must be reversed in direction and increased
to Hd, the coercive force.� (Beiser, 1986)
• �If the external field is reversed in direction
and increased in strength by reversing the
current, the domains reorient until the sample
is again unmagnetized at point c, where
B=0.� (Serway, 1990)
• ��the magnetization does not return to zero,
but remains (D) not far below its saturation
value; and an appreciable reverse field has to
be applied before it is much reduced again
(E).� [where E corresponds to Hc in Fig. 1
(b), and later]�."the field required to reverse
the magnetization (point E on the graph)
varies�" (Akril et al, 1982)
Materials science and magnetism /
electromagnetism books
• �In order to destroy the magnetization, it is
then necessary to apply a reversed field equal
to the coercive force Hc.� (Anderson et al,
1990)
• "To reduce the magnetisation, B, to zero the
direction of the applied magnetic field must be
reversed and its magnitude increased to a
value Hc." (John, 1983) Note here the symbol
B is confusingly used for the magnetization as
discussed later.
• "If the H field is now reversed, the graph
continues down to R in the saturated case.
This represents the H field required to make
the magnetization zero within a saturation
loop and is termed the coercivity of the
material." (Compton, 1986)
• "� the value of H when B=0 is called the
coercivity, Hc; � It follows that the coercivity
Hc is a measure of the field required to reduce
M to zero." (Dugdale, 1993)
• �Note that an external field of strength �Hc,
called the coercive field, is needed to obtain a
microstructure with an equal volume fraction
of domains aligned parallel and antiparallel
to the external field (i.e., B = 0).� (Schaffer et
al, 1999)
Apparently, all the above quotes refer to the
intrinsic coercivity Hci as defined on the M vs H
curve, whereas the B vs H curve was, in fact, used
to explain the properties of the hysteresis loop.
Neither a proper explanation about the validity of
the approximation Hc ≈ Hci nor information on the
type of ferromagnetic materials described by a given
schematic hysteresis loop was provided in all the
quotation cases. Hence, such statements constitute
misconceptions, which could be avoided if the
authors defined the term �coercive force� /
�coercivity� as the reverse field required to
demagnetize (M = 0) the ferromagnetic material
sample with a reference to the M vs H curve.
Otherwise, when referring to the B vs H curve, the
quantity Hc should rather be defined as the field
required to bring the magnetic induction, instead of
the magnetization, to zero. The description in the
text and the curve used in the books cited above,
simply imply that both B and M were equal to zero
at the same value of H, i.e. Hc. However, since B =
µo ( H + M ) , when B = 0, M is equal to -Hc. Only
when Hc is very small, as it is the case for soft
magnetic materials, the approximation M ≈ 0 at B =
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0 and Hc ≈ Hci holds. Without explicitly stating the
necessary conditions for the validity of such
approximation, the presentations of the hysteresis
loop expressed in the above quotes convey an
incorrect concept of the zero magnetization at the
point -Hc on the B vs H curve as applicable to any
kind of ferromagnetic materials.
To predict the value of H on the B vs H curve
for which in fact M = 0, we consider M = B/µo � H.
In the second quadrant of the hysteresis loop (see
Fig. 1), we have �Hc ≤ H ≤ 0, and hence M
diminishes from M = Br/µo at H = 0 to the nonzero
value at -Hc, i.e. M = -Hc. This means that the
direction of the magnetization is still opposite to
that of the applied field. Further increase of the
negative Hc in the third quadrant on the B vs H
curve yields M = 0 at H = �Hci. This is why the
value of Hci on the M vs H curve is always greater
than that of Hc on the B vs H curve. This
relationship is indicated schematically by a dot (the
point -Hci) in Fig. 1 (b). The values in Table 1 in S
& R (2002) illustrate that for strong permanent
magnets Hci is substantially larger in magnitude than
Hc.
B. Misconceptions concerning the meaning of the
saturation induction Bsat
Apart from the two notions of coercivity, the
term of �saturation induction� is also prone to
confusion. If this term is not defined properly,
various misconceptions may arise. Usually, in most
textbooks the term �saturation� refers to the
�process� and thus the corresponding quantities
exhibit no further change after a certain limit is
reached. For instance, a sponge no longer absorbs
any more water after full �saturation�. Similarly, the
magnetization in ferromagnetic materials does not
change after the saturation point is reached at Hsat
(see, Fig. 1). Since M becomes constant, M = Ms,
further increase of the applied field H no longer
changes the value of the magnetization M, as
represented by the straight dotted line in Fig. 1(a).
However, this is not the case for the induction B.
According to Eq. (1) after the saturation point is
reached at Hsat, B still increases with H. Confusion
may occur if the term �saturation� is used with
respect to the B vs H curve. In this case, the
�saturation induction� Bsat reflects that in the
magnetization saturation process a certain limit has
been reached, denoted by a particular point on the B
vs H curve. But it does not mean that B has
reached a definite limit like in the case of M.
Correct descriptions are found in, e.g. Kittel (1996)
who refers the �saturation induction� to the point on
the B vs H graph at which the magnetization reaches
a certain limit; Hammond (1986): "as H is
increased, B increases less and less. It reaches an
almost constant saturation value". However, the
value of B still increases if H is continuously
applied to the sample after saturation of
magnetization, no matter how small the value of
µoH is as compared with µoM. This distinction
between the properties after saturation of the M vs
H curve and those of the B vs H curve is often
misrepresented as shown in Fig. 2 (see, e.g. Pollock
(1990, 1985), Compton (1986), Flinn & Trojan
(1990)). The shape of the B vs H curves apparently
resembles closely the shape of the M vs H curve
with a (nearly) straight horizontal line after
saturation.
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Fig. 2. Hysteresis loop for a ferromagnetic material with the saturation point indicated (adapted from Flinn & Trojan, 1990).
The misconception conveyed by such diagrams
as in Fig. 2 is that after saturation, even if H
increases further, the induction behaves in the same
way as the magnetization, i.e. B = Bs = const as M =
Ms = const. Such misconception is evident in a
number of texts, for instance, "With further increase
in field strength, the magnitude of induction levels
off at a saturation induction, Bs." (Shackelford,
1996), �This maximum value of B is the saturation
flux density Bs� (Callister, 1994), "Bmax is the
maximum magnetic induction�" (Jastrzebski,
1987). The descriptions used in several other
textbooks also reflect similar incorrect interpretation
of the saturation induction Bsat, see, e.g. Pollock
(1990, 1985), Compton (1986), Flinn & Trojan
(1990), Van Vlack (1982), Selleck (1991), Harris &
Hemmerling (1980), Arfken et al (1984), Knoepfel
(2000), Brick et al (1977), and John (1983).
Besides, in some surveyed books, the saturation is
indicated incorrectly on the B vs H curve, e.g.
Buckwalter et al (1987), Whelan and Hodgson
(1982), Brown et al (1995). Those misconceptions
could be avoided if a proper clarification is
provided. It is then necessary to mention that, in
fact, the contribution of the H term to B in Eq. (1)
can be neglected but only for soft magnetic
materials as they become saturated at small values
of H. Some authors Ralls et al (1976), Burke
(1986), Cullity (1972), Anderson et al (1990), Van
Vlack (1970) have explicitly adapted this point of
view. The term �saturation induction� is then used
either under the assumption that after saturation of
the magnetization H contributes to B in a negligible
way, see, e.g. Ralls et al (1976), Anderson et al
(1990), Van Vlack (1970), or it is not worth to
increase B in the actual practice, see, e.g. Burke
(1986). The former is true only for soft magnetic
materials. However, the situation is quite different
for hard magnetic materials for which, as it can be
seen from Table 1 in S & R (2002), the values of
coercivity Hc (in kOe) are close to those of the
remanence Br (in kGs), whereas the intrinsic
coercivity Hci (in kOe) is greater than Br up to three
times.
C. Misconceptions concerning the actual
inclination of the B vs H and/or M vs H
curve after saturation
The misconceptions of this category, closely
related to the category B, concern both the B vs H
graphs and the M vs H graphs. Misconceptions may
arise concerning the actual inclination if on the B vs
H graphs the shape of the hysteresis loops resembles
closely the shape of the M vs H loops and the B-
lines after saturation appear to be represented by a
straight horizontal line with zero inclination. If no
proper explanation is provided the apparent zero
inclination may be taken as a general feature of both
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graphs applicable to all magnetic materials. The
opposite cases arise if on the M vs H graphs the
shape of the hysteresis loops resembles closely the
shape of the B vs H loops and the B-lines after
saturation appear to be represented by lines with a
noticeable inclination. Such cases amount to
mixing up the M vs H graphs with the B vs H
graphs and constitute misconceptions concerning
the features of the M vs H hysteresis loops. Several
cases of both versions of the misconceptions of this
category have been revealed by considering the
shape, inclination, and description of the B vs H and
M vs H graphs in the textbooks.
The misconceptions concerning the B vs H
graphs arise from the neglect of the difference
between the actual and apparent inclination of the
B-lines after saturation. After the magnetization
saturation is reached, M becomes constant: M = Ms.
Thus in the CGS units a further increase of H by 1
Oersted increases B by 1 Gauss, whereas in the SI
units, correspondingly, 1 A/m of H contributes 4π x
10-7 Tesla (i.e. the value of µo) to B. Hence, no
matter which units are used for the y- and x-axis, B
is exactly proportional to H and must be represented
by a straight line. However, the appearance of a
graph depends on the actual inclination of the B-line
after saturation, which is determined by the unit
elements chosen for the y-axis (yunit) and x-axis
(xunit), i.e. the scale used for the graph. To illustrate
how the extension line of the B vs H hysteresis loop
after saturation would look like for different scales,
we have simulated the inclination corresponding to
various scales used for the x- and y-axis as shown in
Fig. 3. The lines S1 to S6 represent the unit element
of the y-axis (yunit) diminished by a factor of 1, 2, 5,
10, 100 and 1000 times, respectively. Thus the ratio
S = yunit : xunit (i.e. the re-scaling factor) equal to 1,
0.5, 0.2, 0.1, 0.01, and 0.001 yields the inclination
45o, 26.6o, 11.3o, 5.7o, 0.57o, and 0.057o for the lines
S1, S2, S3, S4, S5, and S6, respectively. The same
re-scaling factors apply if equivalently the unit
element of the x-axis (xunit) is increased. This is the
case of the graphs where on the x-axis instead of H
the quantity Bo = µoH is used. Then the units of
Tesla are used on both the x- and y-axis, however,
the typical values of Bo are very small as shown in
the second part of Table 1 below.
Fig. 3. Inclination of the B-line on the B (y) vs H
(x) graph after magnetization saturation for various scales. The re-scaling factor: S1 (1), S2 (0.5), S3 (0.2) S4 (0.1), S5 (0.01), and S6 (0.001) corresponds to the inclination, i.e. the angle between the B-line and the x-axis, equal to: 45°, 26.6o, 11.3o, 5.7°, 0.57°, and 0.057°, respectively.
10
In order to plot a 'usable' graph B vs H (M vs H)
at least the first quadrant of hysteresis loop
indicating the two points Br (Mr) and Hc (Hci) must
fit into a standard textbook page. Hence the size of
12
the graph is approximately given by the values of Br
(Mr) and Hc (Hci) and thus different re-scaling
factors are required for different materials. An
approximate value of the suitable re-scaling factor
can be obtained by calculating the ratio Br / Hc
(CGS) or Br / µo Hc (SI) in the given standard units.
This method works well for the narrow and straight
hysteresis loops (soft materials) and the wide ones
(hard materials) for which the values of Br (Mr) are
not too different from their 'saturation' values. For a
slanted hysteresis loop, like in Fig. 4 - the B type,
with the values of Br (Mr) much smaller than their
'saturation' values a multiplicative factor of between
2 to 6 can be applied to Br (Mr), or alternatively the
values of Bs (Ms) may be used, if available. Let us
illustrate the effect of the re-scaling factors by
adopting the unit elements for the y-axis (yunit) and
x-axis (xunit) of equal length, say e.g. one centimeter.
Thus if the ratio Br / Hc (CGS) is, e.g. of the order
of (i) 103 or (ii) 104, the suitable re-scaling factor for
the graph would be (i) 0.001 or (ii) 0.0001. Such
graphs without re-scaling, i.e. using the 1 : 1 unit
labeling on the y- and x-axis, would require the
maximum on the y-axis, Ymax, of not less than (i) 10
m - the height of an average four-storey building or
(ii) 100 m - one-third of the height of Eiffel Tower
in Paris. On such graphs the actual inclination of
the B-lines after saturation would be exactly 45o.
The only drawback would be that they could not be
fitted into any textbook. By squeezing the graphs
along the y-axis (i) 1000 or (ii) 10000 times, a
'usable' size of the graph is obtained. BUT then the
corresponding apparent inclination almost vanishes
to (i) 0.057o - as for the S6 line in Fig. 3 or (ii)
0.0057o - which cannot be discernibly indicated in
Fig. 3. However, the apparent nearly zero
inclination of the B-lines after saturation, e.g. of the
type S5 and S6 in Fig. 3, should not be confused
with the exactly zero inclination of the M-lines after
saturation independent of the scale used.
(a) (b)
Fig. 4. (a) Two extreme cases of the dependence of magnetization for a soft magnetic material on the angle between the applied field H and the easy magnetization direction: the loop (A) - H parallel to the easy direction and (B) - H perpendicular to the easy direction (adapted from Jakubovics, 1994); (b) Modified loop (B) indicating the apparent saturation ap
satM and the full magnetization saturation at the level of the loop (A).
It is worthwhile to analyse some real data and
determine the expected inclination of the B vs H
line after the magnetization saturation. This will
help (a) clarifying the distinction between the hard
and soft magnets in this respect and (b) illustrate the
difference between the actual and apparent
inclination. Let us consider the re-scaling factors
required to fit the B vs H graphs into a textbook
page as described above for three data sets.
Firstly, we consider the materials listed in Table
1 (S & R, 2002). To draw a hysteresis graph one
may adopt the ratio S = yunit : xunit between 1 and
0.5, since the data points (Hci, Hc and Br) are of the
same order of magnitude. Hence, the expected
inclination of the B-lines after saturation should be
noticeable - between 45o and 26.6o corresponding to
13
a straight line in-between the lines S1 and S2 in Fig.
3.
Secondly, we analyse the textbook B vs H
hysteresis diagrams on which the unit elements are
indicated. The pertinent data extracted from
textbooks are listed in Table 1 together with the
values estimated by us. The numerical data for the
magnetic materials listed include: (i) the maximum
values represented on the x- and y-axis denoted as
Xmax and Ymax, respectively, (ii) the values of Br and
Hc read out approximately from the graphs, (iii) the
ratio Br / Hc, and (iv) the approximate inclination
suitable for a given graph. It turns out that in most
hysteresis diagrams the inclination of the B-line
after saturation should be very small (Table 1). Two
exceptions concern Fig. 21.3 (c) in Flinn & Trojan
(1990) and Fig. 29.29 (b) in Lea & Burke (1997),
where the data for a hard magnetic material should
yield a noticeable inclination of about 14o and 17o,
respectively.
Thirdly, since almost all data in Table 1 pertain
to the soft materials, the data for the hard materials
from Table 17.2 in Hummel (1993) are considered.
The results collected in Table 2 show convincingly
that for the hard materials an appreciable inclination
of the B-line after saturation should appear on the B
vs H graphs if plotted to fit into a textbook page.
From the above analysis it is evident that for the
strong permanent magnets (for references, see S &
R, 2002; Hummel, 1993) the B vs H curve no longer
'levels off' after the magnetization saturation.
However, in a number of textbooks the B vs H
hysteresis loop resembles the "M vs H" type curve
with zero inclination (see Section B above).
Generally, no mention is made on the dependence of
the shape and the inclination the B-lines after
saturation on the scale used, which is specific for the
soft and hard magnets. In view of the results in
Table 1, such presentation may be justifiable in the
case of the older books keeping in mind the data
available at that time: see, e.g. Tilley (1976),
Williams et al (1976), Hudson & Nelson (1982),
Sears et al (1982), Bueche (1986; using the B vs
µoH graph), Laud (1987). However, in more recent
books it is rather out-dated. Serway et al (1997)
rather inappropriately differentiate between the hard
and soft materials by referring in their Fig. 12.5 to a
'wide' hysteresis curve with zero inclination of the
B-lines after saturation and to a 'very narrow'
hysteresis curve with noticeable inclination,
respectively.
Concerning the second version of the category
C misconceptions, let us first note that valid cases of
a noticeable inclination may appear on the M vs H
graphs for strongly anisotropic magnetic materials
as discussed briefly in the section D below. For a
full discussion of these cases and references, see S
& R (2002). One has to keep in mind that this
apparent 'inclination' applies only to the range
between the easy axis saturation and the full
saturation (see Fig. 4 (b)). This is distinct from the
cases of the schematic M vs H graphs with the shape
of the hysteresis loops resembling closely the shape
of the B vs H loops and the M-lines after saturation
with a noticeable inclination, like in Fig. 1(b). Such
inappropriate M vs H graphs occur in a few
textbooks: Omar (1975) - most pronounced case,
Keer (1993) - the Ms dotted line in Fig. 5.11 is
indicated incorrectly with non-zero inclination but
the description in text is correct, Halliday et al
(1992) - slight non-zero inclination while Ms not
indicated. These cases cannot be justified by the
anisotropic properties of materials since no proper
explanations are provided by the authors.
14
Table 2. Characteristics of the hysteresis loop for some permanent magnets; the ratios Br (remanence) / Hc (coercivity) or the equivalent ones are given as the order of magnitude only; Br and Hc values are taken from Table 17.2 of Hummel (1993); type of the inclination after saturation refers to the lines in Fig. 3.
Br [kG] Hc [Oe] Br / Hc Br [T] Hc [A/m] Br / µoHc Inclination type Material name 3.95 2400 1.6 0.4 1.9 x 105 1.7 S1 Ba-ferrite ( BaO.6Fe2O3 )
6.45 4300 1.5 0.6 3.4 x 105 1.4 S1 PtCo (77 Pt, 24 Co ) 13 14000 0.9 1.3 1.1 x 106 0.9 S1 Iron-Neodymium-Boron
( Fe14Nd2B1 )
9 51 176 0.9 4 x 103 179 S5 Steel ( Fe-1%C ) 13.1 700 19 1.3 5.6 x 104 18 S4 ~ S5 Alnico 5 DG
( 8 Al, 15 Ni, 24 Co, 3 Cu, 50 Fe )
10 450 22 1 3.6 x 104 22 S4 ~ S5 Vically 2 ( 13V, 52 Co, 35 Fe)
D. Misconceptions concerning the dependence of
the shape of the hysteresis loops on the
direction of the applied field
Another source of confusion may arise due to
the fact that the values of Br and Hc for the same
material may be noticeably different for different
physical conditions to which a given material may
be subjected. Even the same chemically material
may be behave either as a magnetically soft or hard
material, depending on the physical conditions
applied. These aspects will be discussed in detail in
S & R (2002). Here let us consider the possible
different shapes of the hysteresis loop as illustrated
in Fig. 4 for soft materials. According to Jakubovics
(1994) the M vs H loops (A) and (B) in Fig. 4 (a)
are for the same material, but the loop (A)
corresponds to a greater permeability and saturation
value Ms than the loop (B). A curious student
encountering in various books one of the two
distinct types of the hysteresis loop, i.e. in one book
the A-type graph and in another book the B-type
graph - each supposedly for the soft materials,
would certainly be puzzled if no proper explanation
on the physical reasons of such difference is
provided. This unfortunately is the case in some
textbooks (see below). In fact, on either graph: B vs
H or M vs H, the loop (A) is obtained if the field H
is applied parallel to the easy direction (ED) of
magnetization in the material, whereas the loop (B)
is obtained if the field H is applied either parallel to
the hard direction (HD) of magnetization or
perpendicular to the easy direction (see, e.g.,
Cullity, 1972; Jakubovics, 1994; den Broeder &
Draaisma, 1987; Babkair & Grundy, 1987). One
must be careful not to confuse the meaning of
'parallel' and ' perpendicular' directions, since in
some materials the easy direction is perpendicular
to the film surface, and thus the notation used on the
graphs: ⊥ and - referred to the film surface, may
be easily confused with the case: H ED (S & R,
2002).
It appears from our text survey that the
hysteresis loops, both the type: B vs H and M vs H,
for magnetically soft materials are usually presented
in a simplified way as either the loop (A) or (B) in
Fig. 4 (a), with the loop (A) being used more often,
15
especially in the less advanced level texts. The
confusing point is that the loops (B) appear on the B
vs H graphs as an illustration of the distinction
between the soft and hard materials represented by a
narrow slanted loop (B) and a wide rather straight
loop (A), respectively, see, e.g., Knoepfel (2000),
John (1983), Tipler (1991), Giancoli (1991, 1989),
Whelan & Hodgson (1982), and Akrill et al (1982).
In view of the possibility of obtaining for soft
materials also the narrow straight loops - like the
loop (A), having no mention about this possibility
arising from the anisotropic properties of the soft
materials, such description amounts to a partial truth
and may lead to confusion. Thus both the (A) and
(B) types of hysteresis loops are physically possible
for soft materials, however, the values of the
remanence corresponding to each loop are markedly
different. Without clearly stating the physical
conditions applicable to each loop, a simplified
description may not be enough for undergraduate or
lower form students to learn properly the properties
of the soft ferromagnetic materials. Comparison of
the two curves appearing separately in various
textbooks may create an incorrect impression
concerning the physical situation applicable to each
case.
An additional misconception arises when the
two types of the hysteresis loop (A) and (B) are
presented on the same M vs H diagram as in Fig. 4
(a) for the same material (see, e.g. Jakubovics,
1994). For the loop (B), at first the apparent
saturation at a lower value apsatH is achieved, which
correspond to a full internal saturation but along the
easy magnetization direction. The full saturation is
achieved only if the field is further increased, which
brings about rotation of the magnetization from the
easy direction to the direction of the field, which is
completed at BsatH . This then corresponds to the full
saturation of the magnetization at the same level as
for the loop (A) as illustrated in the modified Fig. 4
(b). The apparent saturation may correspond to 70
% of the full saturation (see, e.g. Giancoli (1991))
and fields BsatH of up to several times higher than
AsatH are required to achieve the full saturation Ms.
A survey of research papers (S & R, 2002) reveals
several examples of the modified hysteresis loop (B)
shown in Fig. 4 (b). It turns out that in the
experimental practice the full saturation with the
field in the hard magnetization direction is rarely
achieved. Hence the experimental loops often
indicate an apparent inclination after apsatH similar
to the B-line inclination on the B vs H graphs
discussed in Section C. These two physically
distinct cases should not be confused each with
other. For a more detailed consideration and
discussion of other factors affecting the shape of the
hysteresis loop, which is beyond the scope of this
paper, see, S & R (2002).
E. Misconceptions arising from the hysteresis
loops for both soft and hard materials
presented in the same figure or using the
same scale
Hysteresis loops for both soft and hard magnetic
materials are also found in some texts plotted
schematically for comparison either in the same
diagram (see, e.g. Murray, 1993) as in Fig. 5 or in
two related diagrams using the same scale (see, e.g.
Chalmers, 1982 - see also Section F; Budinski,
1996; Budinski & Budinski, 1999). Normally,
16
such presentation, if physically valid, may help to
convey better a clear picture to students. However,
an opposite effect is achieved, if there exists a great
difference in the scale for the two curves plotted in
this way, as, e.g. in Fig. 5. If these differences are
mentioned neither in the text nor in the figure
caption, the diagrams like in Fig. 5 give rather a
wrong impression to students. In 1949 Kittel (as
quoted by Livingston (1987)) reported that the
values of Hci for the hardest and softest
ferromagnetic materials differ by a factor of 5 x 106
(in the SI units). In the recent decades, the range of
this difference has grown up to 108 (Livingston,
1987). In fact, comparison of the data in Table 1
and 2 reveals that the values of Hc for the hard and
soft magnetic materials listed therein differ by from
several hundreds times in the CGS units (or 3 orders
of magnitude in the SI units) to 10 thousands times
or more (or 6 orders of magnitude in the SI units).
However, it appears from figures like Fig. 5 that the
coercive force Hc for hard magnetic materials is just
only several times larger than that for the soft ones.
This kind of misleading comparison appear, e.g. in
the textbooks by Arfken et al (1984), Brown et al
(1995), Callister (1994), Geddes (1985), John
(1983), Murray (1993), Nelkon & Parker (1978),
Ralls et al (1976), Schaffer et al (1999), Shackelford
(1996), Smith (1993 - note that mixed units are
indicated in Fig. 15.21(c): SI units for the y-axis,
whereas CGS units for the x-axis), Turton (2000),
Whelan & Hodgson (1982).
Fig. 5. The hysteresis loop for soft and hard magnetic materials plotted in the same graph for comparison (adapted from Murray, 1993).
F. Other problems concerning terminology
Various minor problems concerning confusing
terminology have also been identified in our survey.
Table 1 indicates as wide variety of naming
conventions and symbols used. A greater uniformity
in this respect, i.e. adherence to the standard
nomenclature and units, would help avoiding
confusion. Students may easily confuse the various
terms used for the applied field H and magnetic
induction B. As Table 1 indicates other names used
are, e.g., for H: current, magnetic intensity,
magnetic field strength, magnetizing field, while for
B: magnetic field, flux density, total or internal
magnetic field. Chalmers (1982) uses the intensity
of magnetization I, which is defined only in the SI
(Kennelly) unit system (Jiles, 1991). We have
carried out a quick survey among several physics
MPhil students at CityU asking them to identify the
term (i.e. magnetic field, magnetic field, and
magnetic field strength) corresponding to H and B.
As we expected, they can distinguish between the
physical meaning of H and B, but they rather mix up
the corresponding names. To a certain extent, this
17
finding is a reflection of the unhealthy situation
prevailing in the textbooks (see Table 1). The
existing variety of conventions used for the
quantities H, B and M hampers the understanding of
physics, especially even more seriously for the
lower form students.
The improper use of the term 'polarization' to
interpret the B vs H curve occurs in Abele (1993).
Normally, �polarization� is used to describe the
electric quantities rather than the magnetic ones.
�Polarization� of dielectric materials is an analogue
quantity to �magnetization� in magnetism, however,
the two terms are not equivalent. Besides, John
(1983) denotes on the y-axis the magnetization
confusingly as B, �magnetization induced B�, and
describes the x-axis as: �magnetizing force H�.
Improper use of symbols leads to confusion like,
e.g., "saturation magnetization� denoted as �Bs" by
Murray (1993). Another confusing notion is used by
Granet (1980): "magnetization current", which
means the electric current, which induces H, which,
in turn, magnetizes the sample. Such terminology
mixing up the magnetic and electrical notions and/or
quantities may be confusing for students and should
be avoided for pedagogical reasons. Another
example of confusion is: "If the field intensity is
increased to its maximum in this direction, then
reversed again �� (Thornton & Colangelo, 1985).
In fact, there is no �maximum limit� for the applied
field, apart from the limits imposed by the
experimental equipment.
4. A survey of students� understanding of the
hysteresis loop
The effect of the confusion and misconceptions
existing in textbooks on the students� understanding
of the hysteresis loop can be assessed by analysing
the results of examinations or tests. In our lecture
notes for the condensed matter physics course
(Rudowicz, 2000; available from CZR upon
request) we have presented clearly, so briefly, the
distinction between the pertinent notions as well as
warned students about the misinterpretations
discussed in Section A above. However, the analysis
of the results of the CMP examination carried out in
May 2001 indicates that the message has not
reached some students. Eleven students out of total
15 attempted the two questions concerning the
hysteresis loop, stated as follows:
a) Draw a schematic diagram of the magnetic
field intensity inside a material B versus the
(external) applied magnetic field strength H
for an initially unmagnetized ferromagnetic
material.
b) Define the following quantities: (1) the
saturation magnetic field, (2) the remanence,
and (3) the coercive force. Indicate these
quantities on the diagram B versus H (the
hysteresis loop).
Most of them performed not too well since they
often misinterpreted the characteristics of the
hysteresis loop. Common mistakes include, e.g., (a)
mixing up the coercivity with the remanence, (b)
indicating a 'maximum' value of the applied field H
and the magnetic induction B, (c) stating that both B
and M are equal to zero at Hc. Obviously, some of
these misconceptions concerning the hysteresis loop
are quite close to the ones existing in the surveyed
textbooks as discussed above. However, such
misconceptions by our students may not originate
from any insufficient clarifications of the major
18
terms in the books they might have used, but rather
are due to the students� attitudes to learning. In
general, a good interpretation of a particular topic in
textbooks may help teachers to increase the
efficiency of teaching (and save their time which
would have been used for clarifications), whereas
students to improve their understanding of the
physical concepts beyond the level presented at the
lectures. On the contrary, the improper definitions
of the crucial terms and/or outright misconceptions
will most certainly hamper the students�
understanding and may contribute to a reduced
interest in further physics studies, especially at a
lower level of students' education (Hubisz, 2000).
5. Conclusions and suggestions
It appears that the two possible ways of
presenting the hysteresis loop for ferromagnetic
materials, B vs H and M vs H, are, to a certain
extent, confused each with other in several
textbooks. This leads to various misconceptions
concerning the meaning of the physical quantities as
well as the characteristic features of the hysteresis
loop for the soft and hard magnetic materials. We
suggest that the name �coercive force� (or
�coercivity�) and the symbol Hc , correctly defined
for the B vs H curve, should not be used if referred
to the M vs H curve. Using in the latter case the
adjective 'intrinsic' and the symbol Hci is strongly
recommended. It may help avoiding the
misconceptions discussed above and reduce the
present confusion widely spread in the textbooks.
Hence the authors and editors should pay more
attention to proper definitions of the terms involved.
Interestingly, among the books by Beiser (1986,
1991, 1992), the book (1986) belongs to the
�misinterpretation sample�, while the two later
books (1991, 1992) are correct in this aspect. It is
hoped that by bringing the problems in questions to
the attention of physics teachers and students, the
correct interpretation of the hysteresis loop will
prevail in future.
Our survey of textbooks reveal several deeper
pedagogical issues related to the presentation of the
hysteresis loop, which may apply to various other
topics as well. One is the distinction between the
�exact� and �approximate� quantities and the related
description of a physical situation. In the present
case we have considered the approximation �H
small as compared with M�, leading to B ≈ µoM and
Hc ≈ Hci for soft magnetic materials. If the
conditions for which a given approximation is valid
are not clearly stated, the approximate picture may
be implicitly taken as a representation of the exact
situation. The consequences of such misleading
approach may be wide-ranging - from imprinting
misconceptions, i.e. false images, in the students�
minds to misinterpretation of the properties of one
class of materials (here, soft magnets) as being
equivalent to those of another class (here, hard
magnets).
The inherent danger in using �schematic�
diagrams for presentation of the dependencies
between various physical quantities is another
important issue. Having no units and values
provided for the y- and x-axis constitutes a
detachment from a real physical situation. It may
not only hamper students� understanding of the
underlying physics, but also lead to false
impressions about the relationships between the
quantities involved and, in consequence, create
misconceptions. This is best exemplified for the
present topic by Fig. 4 and Fig. 5. The drawbacks of
19
�schematic� representation of each hysteresis loops
are compounded by the �space saving� and using a
combined diagram like in Fig. 5, which implies the
same limits and values are applicable for both types
of magnetic materials. As we amply illustrated
above this is far from the true situation. Schematic
diagrams which do not reflect correctly the
underlying physical situation become a piece of
graphic art only. Providing neither symbols nor
description of the quantities on the x- and y-axis of a
graph (see, e.g. Fig. 15.9 in Machlup, 1988) should
also be avoided in physics text as an inappropriate
from both scientific and pedagogical point of view.
Finally, let us mention the idea of creating a
website listing errors and misconceptions in
textbooks. The individual lecturers could add up
their knowledge in this respect to a well organized
structure listing various topics. We have had
preliminary talks within CityU about the idea of
having such a website residing on the CityU
computer network but the response was rather
muted. Our initial Internet search for the keywords:
'errors', 'misprints', 'corrigenda', 'errata', has,
however, revealed, no relevant sites. A similar idea
was proposed by Hubisz (2000) concerning science
textbooks. Interestingly we have located this
website due to letter in American Physical Society
Newsletter (April, 2001, p.4). Since the URL
address was misprinted, we have tracked this site
down via the university name (North Carolina State
University). Only recently by chance we have learnt
of the existing website listing errors in physics
textbooks:
(http://www.escape.ca/~dcc/phys/errors.html). It
appears that the benefits of such website for teachers
and students in improving general understanding of
physics may be substantial.
Acknowledgements This work was supported by the City University
of Hong Kong through the research grant: QEF #
8710126.
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Dugdale D 1993 Essential of electromagnetism (New York: American Institute of Physics) p 198-199 Elliott R J and Gibson A F 1978 An Introduction to Solid State Physics and its Applications (London: English
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Prentice Hall) p 947-948. Flinn R A and Trojan P K 1990 Engineering Materials and Their applications (Dallas: Houghton Mifflin) p
S178-S182 Geddes S M 1985 Advanced Physics (Houndmills: Macmillan Education) p 57-59 Giancoli D C 1989 Physics for Scientists and Engineers with Modern Physics (New Jersey: Prentice Hall) p
662-663 Giancoli D C 1991 Physics Principles with Applications (New Jersey: Prentice Hall) p 529-531
Granet I 1980 Modern Materials Science (Virginia: Prentice-Hall) p 422-427 Grant I S and Phillips W R 1990 Electromagnetism (Chichester: John Wiley & Sons) p 201-242 Gray H J and Isaacs A 1975 A New Dictionary of Physics (London: Longman) p 268 Halliday D, Resnick R and Krane K S 1992 Physics Vol.2 (New York: John Wiley & Sons) p 814 Hammond P 1986 Electromagnetism for Engineers An Introductory Course (Oxford: Pergamon press) p 134-
137 Harris N C and Hemmerling E M 1980 Introductory Applied Physics (New York: McGraw-Hill) p 570-571 Hickey & Schibeci 1999 Phys. Educ. 34 383-388 Hoon S R and Tanner B K 1985 Phys. Educ. 20 61-65 Hubisz J L 2000 Review of Middle School Physical Science Texts (http://www.psrc-
online.org/curriculum/book.html) - accessed June 2001 p 1-98 Hudson A and Nelson R 1982 University Physics (San Diego: Harcourt Brace Jovanovich) p 669
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Hummel R E 1993 Electronic Properties of Materials (Berlin: Springer-Verlag) p 314, 319 Jakubovics J P 1994 Magnetism and Magnetic Materials (Cambridge: The Institute of Materials) Jastrzebski Z D 1987 The Nature and Properties of Engineering materials (New York: John Wiley & Sons) p
482-485 Jiles D 1991 Introduction to Magnetism and Magnetic Materials (London: Chapman & Hall) p 70-73 John V B 1983 Introduction to Engineering Materials (London: Macmillan) p 130-131 Keer H V 1993 Principles of the Solid State (New York: John Wiley and Sons) p 235-236 Kittel C 1996 Introduction to Solid State Physics (New York: John Wiley & Sons) p 468-470 Knoepfel H E 2000 Magnetic fields A Comprehensive Theoretical Treatise for Practical Use (New York: John
Wiley & Sons) p 486-491 Lapedes D N 1978 Dictionary of Physics and Mathematics (New York: McGraw-Hill) p 469 Laud B B 1987 Electromagnetics (New York: John Wiley & Sons) p 162-163 Lea S M and Burke J R 1997 Physics The Nature of Things (New York: West Publishing) p 941-942 Lerner R G and Trigg G L 1991 Encyclopedia of Physics (New York: VCH Publishers) p 692-693 Levy R A 1968 Principles of Solid State Physics (New York: Academic Press) p 258-260 Livingston 1987 Upper and lower limits of hard and soft magnetic properties in Proceedings of the
International Symposium on Physics of Magnetic Materials ed M Takahashi et al (Singapore: World Scientific) p 3-16
Lord M P 1986 Dictionary of Physics (London: Macmillan) p 140 Lorrain P and Corson D R 1979 Electromagnetism (San Francisco: W.H. Freeman and Company) p 342-345 Lovell M C, Avery A J and Vernon M W 1981 Physical Properties of Materials (New York: Van Nostrand
Reinhold) p 189 Machlup S 1988 Physics (New York: John Wiley & Sons) p 459 Meyers R A 1990 Encyclopedia of Modern Physics (San Diego: Harcourt Brace Jovanovich) p 254-255 Murray G T 1993 Introduction to Engineering Materials Behavior, Properties, and Selection (New York:
Marel Dekker) p 529-531 Nelkon M and Parker P 1978 Advanced Level Physics (London: Heinemann Educational Books) p 843 Omar M A 1975 Elementary Solid State Physics: Principles and Applications (Massachusetts: Addison-
Wesley) p 461 Ouseph P J 1986 Technical Physics (New York: Delmar) p 537-538 Parker S P 1993 Encyclopedia of Physics (New York: McGraw-Hill) p 733 Pitt V H 1986 The Penguin Dictionary of Physics (Middlesex: Penguin books) p 186-187 Pollock D D 1985 Physical of Materials for Engineers Vol. II (Boca Raton: CRC Press) p 138-139 Pollock D D 1990 Physics of Engineering Materials (New Jersey: Prentice Hall) p 587-589 Ralls K M, Courtney T H and Wulff 1976 Introduction to Materials Science and Engineering (New York: John
Wiley & Sons) p 575-577 Rhyne J J 1983 Magnetic Materials in Concise Encyclopedia of Solid State Physics ed R G Lerner & G L
Trigg (London: Addison-Wesley Publishing) p 160-162 Rogalski M S and Palmer S B 2000 Solid-State Physics (Australia: Gordon and Breach Science Publishers) p
379 Rudowicz C, 2001, Lecture Notes: Condensed Matter Physics, City University of Hong Kong, unpublished. Schaffer J P, Saxena a, Antolovich S D, Sanders Jr. T H and Warner S B 1999 The Science and Design of
Engineering Materials (Boston: McGraw-Hill) p 527-530 Sears F W, Zemansky M W and Young H D 1982 University Physics (California: Addison-Wesley) p 673-674 Selleck E 1991 Technical Physics (New York: Delmar) p 879-880 Serway R A 1990 Physics for Scientists & Engineers with Modern Physics (Philadelphia: Saunders College) p
857-859 Serway R A, Moses C J and Moyer C A 1997 Modern Physics (Fort: Saunders College) p 481-483 Shackelford J F 1996 Introduction to Materials Science for Engineers (New Jersey: Prentice Hall) p 507-512 Skomski R and Coey J M D 1999 Studies in Condensed Matter Physics Permanent Magnetism (Bristol:
Institute of Physics) p 169-174 Smith W F 1993 Foundations of Materials Science and Engineering (New York: McGraw-Hill) p 827-828 Sung H W F and Rudowicz C 2002 J. Mag. Magn. Mat. - submitted Feb 2002
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Thornton P A and Colangelo V J 1985 Fundamentals of Engineering Materials (New Jersey: Prentice-Hall) p 372-373
Tilley D E 1976 University Physics for Science and Engineering (California: Cummings Publishing) p 532-534Tipler P A 1991 Physics for Scientists and Engineers (New York: Worth) p 886-888 Turton R 2000 The Physics of Solids (Oxford: Oxford University Press) p 237 Van Vlack L H 1970 Materials Science for Engineers (Menlo Park: Addison-Wesley) p 327-328 Van Vlack L H 1982 Materials for Engineering: Concepts and Applications (Menlo Park: Addison-Wesley) p
544 Vermariën H, McConnell E and Li Y F 1999 Reading / recording devices The Measurement, Instrumentation,
and Sensors Handbook ed J G Webster (Florida: CRC Press) p 96-24-25 Wert C A and Thomson R M 1970 Physics of Solids (New York: McGraw-Hill) p 455-456 Whelan P M and Hodgson M J 1982 Essential Principles of Physics (London: John Murray) p 429 Williams J E Trinklein F E and Metcalfe H C 1976 Modern Physics (New York: Holt, Rinehart and Winston) p
468 Wilson I 1979 Engineering Solids (London: McGraw Hill) p 119-123
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Appendix I. List of other surveyed textbooks not included in the references The list of textbooks surveyed, in which no relevant misconceptions and/or confusions were identified and
which are not quoted in text, is given below.
Benson H 1991 University Physics (New York: John Wiley & Sons) p 653 Blakemore J S 1985 Solid State Physics (Cambridge: Cambridge University Press) p 450 Bube R H 1992 Electrons in Solids An Introductory Survey (Boston: Academic Press) p 261-262 Burns G 1985 Solid State Physics (Orlando: Academic Press) p 614 Christman J R 1988 Fundamentals of Solid State Physics (New York: John Wiley & Sons) p 369 Coren R L 1989 Basic Engineering Electromagnetics An Applied Approach (New Jersey: Prentice Hall) p 76-
77 Craik D 1995 Magnetism Principles and Applications p 105 Crangle J 1991 Solid State Magnetism (London: Edward Arnold) p 168-171 Dekker A J 1960 Solid State Physics (London: Macmillan & Co) p 476 Enz C P 1992 Lecture Notes in Physics vol. 11 A Course on Many-Body Theory Applied to Solid-State Physics
(Singopore: World Scientific) p 267 Feynman Leighton and Sands 1989 Commemorative Issue The Feynman Lectures on Physics Vol. II (Addison-
Wesley: California) p 36-5-36-9 Gershenfeld N 2000 The Physics of Information Technology (Cambridge: Cambridge University Press) p 193 Guinier A and Jullien R 1989 The Solid State From Superconductors to Superalloys (Oxford: Oxford
University Press) p 163-166 Halliday D, and Resnick R 1978 Physics Part I and II (New York: John Wiley & Sons) p 827 Hammond P and Sykulski J K 1994 Engineering Electromagnetism Physical Processes and Computation
(Oxford: Oxford University Press) p 223-225 Hook J R and Hall H E 1991 Solid State Physics (Chichester: John Wiley & Sons) p 251 Joseph A, Pomeranz K, Prince J and Sacher D 1978 Physics for Engineering Technology (New York: John
Wiley & Sons) p 442 Kahn O 1999 Magnetic anisotropy in molecule-based magnets in Metal-Organic and Organic Molecular
Magnets ed P Day and A E Underhill (Cambridge: Royal Society of Chemistry) p 150-168 Kinoshita M 1999 Molecular-based magnets: setting the scene in Metal-Organic and Organic Molecular
Magnets ed P Day and A E Underhill (Cambridge: Royal Society of Chemistry) p 4-21 Lorrain P and Corson D R 1979 Electromagnetism (San Francisco: W.H. Freeman and Company) p 342-345 Marion J B and Hornyak W F 1984 Principles of Physics (Philadelphia: Saunders College Publishing) p 750 Myers H P 1991 Introductory Solid State Physics (London: Taylor & Francis) p 377 Narang B S 1983 Material Science and Processes (Delhi: CBS) p 66-67 Ohanian H C 1989 Physics Vol.2 (New York: W.W. Norton) p 818 Parker S P 1988 Solid-State Physics Source Book (New York: Mcgraw-Hill) p 223-230 Radin S H and Folk R T 1982 Physics for Scientists and Engineers (New Jersey: Prentice-Hall) p 657 Rosenberg H M 1983 The Sold State (Oxford: Clarendon) p 202 Rudden M N and Wilson J 1993 Elements of Solid State Physics (Chichester: John Wiley & Sons) p 103, 109 Schneider, Jr S J, Davis J R, Davidson G M, Lampman S R, Woods M S and Zorc T B 1991 Engineered
Materials Handbook Vol. 4 (USA: ASM International) p 1162 Swartz C E 1981 Phenomenal Physics (New York: John Wiley & Sons) p 609 Tanner B K 1995 Introduction to the Physics Electrons in Solids (Cambridge: Cambridge University Press) p
181-182 Tippens P E 1991 Physics (New York: Glencoe/McGraw-Hill) p 671-672 Van Vlack L H 1989 Elements of Materials Science and Engineering (Menlo park: Addison-Wesley) p 450-453Vonsovskii S V 1974 Magnetism Vol. One (New York: John Wiley & Sons) p 42-43 Young H D and Freeman R A 1996 Extended Version with Modern Physics: University Physics
(Massachusetts: Addison-wesley) p 926 Zafiratos C D 1985 Physics (New York: John Wiley & Sons) p 673-674
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